On Gelfond’s conjecture about the sum of digits of prime numbers
Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 415-422.

Dans cet article nous exposons les étapes importantes de la preuve de la conjecture de Gelfond [6] (1968) dans un travail récent en collaboration avec Christian Mauduit [11] concernant la somme des chiffres des nombres premiers, dans l’esprit de l’exposé donné à Edimbourg dans le cadre des Journées Arithmétiques 2007.

The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.

@article{JTNB_2009__21_2_415_0,
     author = {Jo\"el Rivat},
     title = {On {Gelfond{\textquoteright}s} conjecture about the sum of digits of prime numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {415--422},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {2},
     year = {2009},
     doi = {10.5802/jtnb.678},
     zbl = {pre05620658},
     mrnumber = {2541433},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.678/}
}
Joël Rivat. On Gelfond’s conjecture about the sum of digits of prime numbers. Journal de Théorie des Nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 415-422. doi : 10.5802/jtnb.678. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.678/

[1] C. Dartyge and G. Tenenbaum, Sommes des chiffres de multiples d’entiers. Ann. Inst. Fourier (Grenoble) 55 (2005), 2423–2474. | Numdam | MR 2207389 | Zbl 1110.11025

[2] E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presques premiers. Mathematische Annalen 305 (1996), 571–599. | MR 1397437 | Zbl 0858.11050

[3] E. Fouvry and C. Mauduit, Méthodes de crible et fonctions sommes des chiffres. Acta Arithmetica 77 (1996), 339–351. | MR 1414514 | Zbl 0869.11073

[4] J. Friedlander and H. Iwaniec, The polynomial X 2 +Y 4 captures its primes. Ann. of Math. (2) 148 (1998), 945–1040. | MR 1670065 | Zbl 0926.11068

[5] J. Friedlander and H. Iwaniec, Asymptotic sieve for primes. Ann. of Math. (2) 148 (1998), 1041–1065. | MR 1670069 | Zbl 0926.11067

[6] A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données. Acta Arithmetica 13 (1968), 259–265. | MR 220693 | Zbl 0155.09003

[7] G. Harman, Primes with preassigned digits. Acta Arith. 125 (2006), 179–185. | MR 2277847 | Zbl pre05082231

[8] D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math. 34 (1982), 1365–1377. | MR 678676 | Zbl 0478.10024

[9] D. R. Heath-Brown, Primes represented by x 3 +2y 3 . Acta Math. 186 (2001), 1–84. | MR 1828372 | Zbl 1007.11055

[10] K. Mahler, The Spectrum of an Array and Its Application to the Study of the Translation Properties of a Simple Class of Arithmetical Functions. II: On the Translation Properties of a Simple Class of Arithmetical Functions. J. of Math. Phys. Mass. Inst. Techn. 6 (1927), 158–163.

[11] C. Mauduit, J. Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers. Annals of Mathematics, à paraître.

[12] I. I. Piatetski-Shapiro, On the distribution of prime numbers in sequences of the form [f(n)]. Mat. Sbornik N.S. 33(75) (1953), 559–566. | MR 59302 | Zbl 0053.02702

[13] W. Sierpiński, Sur les nombres premiers ayant des chiffres initiaux et finals donnés. Acta Arith. 5 (1959), 265–266. | MR 109811 | Zbl 0094.25505

[14] R. C. Vaughan, An elementary method in prime number theory. Acta Arithmetica 37 (1980), 111–115. | MR 598869 | Zbl 0448.10037

[15] I. M. Vinogradov, The method of Trigonometrical Sums in the Theory of Numbers, translated from the Russian, revised and annotated by K.F. Roth and A. Davenport. Interscience, London, 1954. | MR 2104806 | Zbl 0055.27504

[16] D. Wolke, Primes with preassigned digits. Acta Arith. 119 (2005), 201–209. | MR 2167722 | Zbl 1080.11064